# Optimization of microchannel heat sinks using entropy generation minimization method

**ABSTRACT** In this study, an entropy generation minimization (EGM) procedure is employed to optimize the overall performance of microchannel heat sinks. This allows the combined effects of thermal resistance and pressure drop to be assessed simultaneously as the heat sink interacts with the surrounding flow field. New general expressions for the entropy generation rate are developed by considering an appropriate control volume and applying mass, energy, and entropy balances. The effect of channel aspect ratio, fin spacing ratio, heat sink material, Knudsen numbers and accommodation coefficients on the entropy generation rate is investigated in the slip flow region. Analytical/empirical correlations are used for heat transfer and friction coefficients, where the characteristic length is used as the hydraulic diameter of the channel. A parametric study is also performed to show the effects of different design variables on the overall performance of microchannel heat sinks

**0**

**0**

**·**

**1**Bookmark

**·**

**157**Views

- [show abstract] [hide abstract]

**ABSTRACT:**Cooling systems take a significant portion of the total mass and/or volume of power electronic systems. In order to design a converter with high power density, it is necessary to minimize the converter's cooling system volume for a given maximum tolerable thermal resistance. This paper theoretically investigates whether the cooling system volume can be significantly reduced by employing new advanced composite materials like isotropic aluminum/diamond composites or anisotropic highly orientated pyrolytic graphite. Another strategy to improve the power density of the cooling system is to increase the rotating speed and/or the diameter of the fan, which is limited by increasing power consumption of the fan. Fan scaling laws are employed in order to describe volume and thermal resistance of an optimized cooling system (fan plus heat sink), resulting in a single compact equation dependent on just two design parameters. Based on this equation, a deep insight into different design strategies and their general potentials is possible. The theory of the design process is verified experimentally for cooling a 10 kW converter. Further experimental results showing the result of the operation of the optimized heat sink are also presented.IEEE Transactions on Components, Packaging, and Manufacturing Technology 05/2011; · 1.26 Impact Factor - SourceAvailable from: InTech
##### Chapter: Thermodynamic Optimization

03/2012; , ISBN: 978-953-51-0278-6 - SourceAvailable from: Ahmed Mohammed Adham[show abstract] [hide abstract]

**ABSTRACT:**In this paper, the optimization of the cooling performance of a rectangular microchannel heat sink is investigated with four different gaseous coolants; air, ammonia gas, dichlorodifluoromethane (R-12) and chlorofluoromethane (R-22). A systematic robust thermal resistance model together with a methodical pumping power calculation is used to formulate the objective functions, the thermal resistance and pumping power. The non-dominated sorting genetic algorithm (NSGA-II), a multi-objective algorithm, is applied in the optimization procedure. The optimized thermal resistances obtained are 0.178, 0.14, 0.08 and 0.133°K/W for the pumping powers of 6.4, 4, 22.4 and 16.5 W for air, ammonia gas, R-12 and R-22, respectively. These results show that among all the gaseous coolants investigated in the current study, ammonia gas exhibited balanced thermal and hydrodynamic performances. Due to the Montreal Protocol, the coolant R-12 is no longer produced while R-22 will eventually be phased out. The results from ammonia provide a strong motivation to conduct more investigations on the potential usage of this gaseous coolant in the electronic cooling industry.Procedia Engineering. 01/2013; 56:337–343.

Page 1

IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL. 32, NO. 2, JUNE 2009 243

Optimization of Microchannel Heat Sinks Using

Entropy Generation Minimization Method

Waqar Ahmed Khan, J. Richard Culham, Member, IEEE, and M. Michael Yovanovich

Abstract—In this paper, an entropy generation minimization

(EGM) procedure is employed to optimize the overall perfor-

mance of microchannel heat sinks. This allows the combined

effects of thermal resistance and pressure drop to be assessed

simultaneously as the heat sink interacts with the surrounding

flow field. New general expressions for the entropy generation rate

are developed by considering an appropriate control volume and

applying mass, energy, and entropy balances. The effectof channel

aspect ratio, fin spacing ratio, heat sink material, Knudsen num-

bers, and accommodation coefficients on the entropy generation

rate is investigated in the slip flow region. Analytical/empirical

correlations are used for heat transfer and friction coefficients,

where the characteristic length is used as the hydraulic diameter

of the channel. A parametric study is also performed to show the

effects of different design variables on the overall performance of

microchannel heat sinks.

Index Terms—Analytical model, entropy generation minimiza-

tion, microchannel heat sinks, optimization.

NOMENCLATURE

Total heating surface area [mm ].

Cross-section area of a single fin [mm ].

Hydraulic diameter [mm].

Friction factor.

Volume flow rate [m /s].

Equality and inequality constraints.

Channel height [m].

Specific enthalpy of the fluid [J/kg].

Average heat transfer coefficient [W/m K].

Number of imposed constraints.

Knudsen number

Thermal conductivity of solid [W/m K].

Sum of entrance and exit losses.

,

.

Manuscript received March 20, 2006; revised July 26, 2006. Current ver-

sion published July 22, 2009. This work was supported by the Centre for Mi-

croelectronics Assembly and Packaging and Natural Sciences and Engineering

Research Council of Canada. This work was recommended for publication by

Associate Editor B. Sammakia upon evaluation of the reviewers comments.

W. A. Khan is with the Department of Engineering Sciences, National

University of Sciences and Technology, PN Engineering College, PNS Jauhar,

Karachi 75350, Pakistan.

J. R. Culham and M. M. Yovanovich are with the Microelectronics Heat

TransferLaboratory,DepartmentofMechanicalEngineering,UniversityofWa-

terloo, Waterloo, ON N2L 3G1 Canada (e-mail: rix@mhtlab.uwaterloo.ca).

Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCAPT.2009.2022586

Ratio of thermal conductivity of fluid to solid

.

Thermal conductivity of fluid [W/m K].

Lagrangian function.

Length of channel in flow direction [mm].

Fin parameter [m].

Total mass flow rate [kg/s].

Total number of microchannels.

Number of design variables.

Nusselt number based on hydraulic diameter

.

Pressure [Pa].

Peclet number based on hydraulic diameter

.

Prandtl number.

Heat transfer rate from the base [W].

Heat transfer rate from the fin [W].

Heat flux [W/m ].

Resistance [K/W].

Reynolds number based on hydraulic diameter

.

Total entropy generation rate [W/K].

Specific entropy of fluid [J/kg K].

Absolute temperature [K].

Thickness [m].

Average velocity in channels [m/s].

Specific internal energy [J/kg].

Specific volume of fluid [m /kg].

Width of heat sink [mm].

Half of the channel width [mm].

Half of the fin thickness [mm].

Design variables.

GREEK SYMBOLS

Pressure drop across microchannel [Pa].

Thermal diffusivity [m /s] or constant defined by

(23).

Channel aspect ratio

Heat sink aspect ratio

Fin spacing ratio

Ratio of specific heats

.

.

.

.

1521-3331/$25.00 © 2009 IEEE

Page 2

244IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL. 32, NO. 2, JUNE 2009

, Mean free path [m] or Lagrangian multipliers.

Absolute viscosity of fluid [kg/m s].

Kinematic viscosity of fluid [m /s].

Fluid density [kg/m ].

Tangential momentum accommodation coefficient

(TMAC).

Energy accommodation coefficient.

Slip velocity coefficient.

Temperature jump coefficient.

SUBSCRIPTS

Ambient.

Average.

Base surface.

Channel.

Capacity.

Contraction and expansion.

Convective.

Fluid.

Single fin.

Hydraulic.

Heat sink.

Entrance.

Exit.

Slip.

Thermal.

Wall.

I. INTRODUCTION

O

trend in the electronic industry toward denser and more pow-

erful products requires a higher level of performance from

cooling technology. After the pioneering work of Tuckerman

and Pease [1], microchannels have received considerable

attention especially in microelectronics. Microchannel heat

sinks provide a powerful means for dissipating high heat flux

with small allowable temperature rise. Due to an increase

in surface area and a decrease in the convective resistance

at the solid/fluid interface, heat transfer is enhanced in mi-

crochannels. These heat sinks can be applied in many important

fields like microelectronics, aviation and aerospace, medical

treatment, biological engineering, materials sciences, cooling

of high-temperature superconductors, thermal control of film

deposition, and cooling of powerful laser mirrors. The two

important characteristics of microchannel heat sinks are the

high heat transfer coefficients and lower friction factors.

In this paper, an entropy generation minimization (EGM)

criterion is used to determine the overall performance of a

microchannel heat sink, which allows the combined effect of

thermal resistance and pressure drop to be assessed through the

NE of the important aspects of electronics packaging

is the thermal management of electronic devices. The

simultaneous interaction with the heat sink. A general expres-

sion for the entropy generation rate is obtained by using the

conservation equations for mass and energy with the entropy

balance.

II. LITERATURE REVIEW

The concept of “microchannel heat sinks” was first intro-

duced by Tuckerman and Pease [1]. Following this work, sev-

eral experimental, numerical, and theoretical studies on the op-

timization of microchannel heat sinks have been carried out.

These studies are reviewed in this section.

Steinke and Kandlikar [2] presented a comprehensive review

of conventional single-phase heat transfer enhancement tech-

niques. They discussed several passive and active enhancement

techniques for minichannels and microchannels. Some of their

proposed enhancement techniques include fluid additives, sec-

ondary flows, vibrations, and flow pulsations.

Kandlikar and Grande [3] explored the cooling limits of the

plain rectangular microchannels with water cooling for high

heat flux dissipation and illustrated the need for enhanced mi-

crochannels. They described a simplified and well-established

fabrication process to fabricate 3-D microchannels. They also

demonstrated the efficacy of the fabrication process in fabri-

cating complex microstructures within a microchannel.

Knight et al. [4], [5], Perret et al. [6], [7], Kim [8], Upadhye

and Kandilkar [9], Kandlikar and Grande [3], and Liu and

Garimella [10] developed theoretical optimization procedures

to minimize the overall thermal resistance of microchannel heat

sinks for a given pressure drop, whereas Singhal et al. [11], and

Kandlikar and Upadhye [12] carried out analyses to optimize

the channel configuration that yields a minimum pressure drop

for a given heat load. They presented solution procedures for

laminar/turbulent flow and generalized their results in terms of

different heat sink parameters.

Kleiner et al. [13], Aranyosi et al. [14], and Harris et al. [15]

investigated theoretically and experimentally the performance

of microchannel heat sinks. They modeled the performance in

terms of thermal resistance, pressure drop, and pumping power

as a function of heat sink dimensions, tube dimensions, and air

flow rate. Their results show an enhancement in heat removal

capability compared to traditional forced air-cooling schemes.

Garimella and Singhal [16] and Jang and Kim [17] analyzed

experimentally the pumping requirements of microchannel heat

sinks and optimized the size of the microchannels for minimum

pumping requirements. Jang and Kim [17] showed that the

cooling performance of an optimized microchannel heat sink

subject to an impinging jet is enhanced by about 21% compared

to that of the optimized microchannel heat sink with parallel

flow under the fixed-pumping-power condition.

Choquette et al. [18], Zhimin and Fah [19], Meysenc et al.

[20], Chong et al. [21], Liao et al. [22], Ryu et al. [23], Wei and

Joshi[24],andJean-Antoineetal.[25]performednumericalop-

timizations of thermal performance of microchannel heat sinks

for given pumping power/pressure drop. Zhimin and Fah [17]

considered laminar, turbulent, developed, and developing flow

and heat transfer in the analysis. Using self-developed software,

they investigated the effects of heat sink channel aspect ratio,

fin-width-to-channel-width ratio, and the channel width on the

Page 3

KHAN et al.: OPTIMIZATION OF MICROCHANNEL HEAT SINKS USING ENTROPY GENERATION MINIMIZATION METHOD 245

performance of heat sink. They found that the channel width is

the most important parameter and governs the performance of

a microchannel heat sink. Min et al. [26] showed numerically

that the tip clearance can improve the cooling performance of

a microchannel heat sink when tip clearance is smaller than a

channel width. Delsman et al. [27] performed a CFD study for

the optimization of the plate geometry to reach the design target

regarding the quality of the flow distribution.

Haddad et al. [28] investigated numerically the entropy

generation due to steady laminar forced convection fluid flow

through parallel plate microchannels. They discussed the effect

of Knudsen, Reynolds, Prandtl, and Eckert numbers and the

nondimensional temperature difference on entropy generation

within the microchannel. The entropy generation within the

microchannel is found to decrease as Knudsen number in-

creases, and it is found to increase as Reynolds, Prandtl, and

Eckert numbers and the nondimensional temperature difference

increase. The contribution of the viscous dissipation in the total

entropy generation increases as Knudsen number increases

over wide ranges of the flow controlling parameters.

It is obvious from the literature survey that all the studies re-

latedtooptimizationofmicrochannelheatsinksareusedtomin-

imize thermal resistance for a given pressure drop or to min-

imize pumping power for a specified thermal resistance. No

study exists to optimize both thermal and hydraulic resistances

simultaneously. In this study, an EGM method is applied as a

unique measure to study the optimization of thermal and hy-

draulic resistances simultaneously. All relevant design parame-

ters for microchannel heat sinks, including geometric parame-

ters, material properties, and flow conditions are optimized si-

multaneously by minimizing the entropy generation rate

subject to manufacturing and design constraints.

III. MODEL DEVELOPMENT

The geometry of a microchannel heat sink is shown in

Fig. 1(a). The length of the heat sink is

The top surface is insulated and the bottom surface is uniformly

heated. A coolant passes through a number of microchannels

along the -axis and takes heat away from the heat dissipating

electronic component attached below. There are

and each channel has a height

of each fin is

whereas the thickness of the base is

temperature of the channel walls is assumed to be

inlet water temperature of

. At the channel wall, the slip

flow velocity and temperature jump boundary conditions were

applied to calculate friction and heat transfer coefficients.

Taking advantage of symmetry, a control volume is selected

for developing the entropy generation model, as shown in

Fig. 1(b). The length of the control volume is taken as unity for

convenience and the width and height are taken as

and , respectively. This control volume includes half

of the fin and half of the channel along with the base. The side

surfaces AB and CD and the top surface AC of this CV can be

regarded as impermeable, adiabatic, and shear free (no mass

transfer and shear work transfer across these surfaces). The heat

flux over the bottom surface BD of the CV is . The ambient

temperature is

and the surface temperature of the channel

wall is

. The bulk properties of the coolant are represented

and the width is.

channels

and width . The thickness

. The

with an

Fig. 1. Microchannel heat sink: geometry, control volume, and definitions.

by

at the outlet, respectively. The irreversibility of this system

is due to heat transfer across the finite temperature difference

and to friction. Heat transfer in the control volume

[Fig. 1(b)] is a conjugate, combining heat conduction in the fin

and convection to the cooling fluid. To simplify the analysis,

the following assumptions were employed.

1) Uniform heat flux on the bottom surface.

2) Smooth surface of the channel.

3) Adiabatic fin tips.

4) Isotropic material.

5) Fully developed heat and fluid flow.

6) Steady and laminar flow.

7) Slip flow (i.e.,

effects.

8) Incompressible fluid with

properties.

9) Negligible axial conduction in both the fin and fluid.

10) Changes in kinetic and potential energies are negligible.

The mass rate balance for the steady state reduces to

, , and at the inlet and by, , and

) with negligible creep

constant thermophysical

(1)

Assuming negligible changes in the kinetic and potential ener-

gies,thefirstlawofthermodynamicsforasteady-statecondition

can be written as

(2)

Thecombinationofspecificinternalandflowenergiesisdefined

as specific enthalpy; therefore, the energy rate balance reduces

further to

(3)

From the second law of thermodynamics

(4)

Page 4

246 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL. 32, NO. 2, JUNE 2009

For steady state,

from the bottom of the heat sink

rate balance reduces to

, and the total heat transferred

, so the entropy

(5)

where

base.

Integrating Gibb’s equation from inlet to the outlet gives

represents the absolute temperature of the heat sink

(6)

Combining (3), (5), and (6), the entropy generation rate can be

written as

(7)

Rearranging the terms and writing

, we have

(8)

As

, the entropy generation rate can be written as

(9)

where

pressure drop across the channel. This expression describes the

entropy generation rate model completely and it shows that the

entropy generation rate depends on the total thermal resistance

and the pressure drop across the channel, provided that

the heat load and ambient conditions are specified. If

volume flow rate, then the total mass flow rate can be written as

is the total thermal resistance andis the total

is the

(10)

The average velocity in the channel

is given by

(11)

where

is the number of channels given by

(12)

IV. THERMAL RESISTANCE

The total thermal resistance is defined as

(13)

where

fluid temperature. These temperatures are given by

is the heat sink base temperature andis the bulk

(14)

(15)

Assuming

resistances can be written as

, the convective and fluid thermal

(16)

(17)

where

is the total heat surface area and is given by

(18)

with

(19)

(20)

Using slip flow velocity and temperature jump boundary condi-

tions, Khan [29] solved the energy equation and developed the

following theoretical model for the dimensionless heat transfer

coefficient for a parallel plate microchannel:

(21)

where

(22)

(23)

with

(24)

(25)

(26)

(27)

Using (14)–(21), (13) can be written as

(28)

where

(29)

(30)

(31)

Page 5

KHAN et al.: OPTIMIZATION OF MICROCHANNEL HEAT SINKS USING ENTROPY GENERATION MINIMIZATION METHOD 247

with

(32)

(33)

(34)

(35)

(36)

V. PRESSURE DROP

The pressure drop associated with flow across the channel is

given by

(37)

where the friction factor

channel aspect ratio, and slip velocity coefficient [29], and can

be written as

depends on the Reynolds number,

(38)

Kleiner et al. [13] used experimental data from Kays and

London [30] and derived the following empirical correlation

for the entrance and exit losses

and fin thickness:

in terms of channel width

(39)

VI. ENTROPY GENERATION RATE

Substituting (28) and (37) into (9), we get

(40)

(41)

where

to heat transfer and fluid friction, respectively, and

by

and show the entropy generation rates due

is given

(42)

VII. OPTIMIZATION PROCEDURE

The problem considered in this study is to minimize the

entropy generation rate, given by (9) or (40), for the optimal

overall performance of microchannel heat sinks. If

resents the entropy generation rate that is to be minimized

subject to equality constraints

equality constraints

mathematical formulation of the optimization problem may be

written in the following form:

rep-

and in-

, then the complete

(43)

TABLE I

ASSUMED PARAMETER VALUES

subject to the equality constraints

(44)

and inequality constraints

(45)

where

constraints and

and are the imposed equality and inequality

denotes the vector of the design variables

. The objective function can be redefined

by using the Lagrangian function as follows:

(46)

where

positive or negative but the

condition for

consideration, is that the Hessian matrix of

semidefinite, i.e.,

and are the Lagrange multipliers. The

must be

to be a local minimum of the problem, under

can be

. The necessary

should be positive

(47)

Foralocalminimumtobeaglobalminimum,alltheeigenvalues

of the Hessian matrix should be

A system of nonlinear equations is obtained, which can

be solved using numerical methods such as a multivariable

.

Page 6

248 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL. 32, NO. 2, JUNE 2009

TABLE II

RESULTS OF OPTIMIZATION

Newton–Raphson method. This method has been described in

[31] and applied by Culham and Muzychka [32] and Culham

et al. [33] to study the optimization of plate fin heat sinks, and

by Khan et al. [34], [35] for pin fin heat sinks and tube banks.

In this paper, the same approach is used to optimize the overall

performance of a microchannel heat sink in such a manner

that all relevant design conditions combine to produce the best

possible heat sink for the given constraints. The optimized

results are then presented in graphical form.

VIII. CASE STUDIES AND DISCUSSION

Theassumedparametervalues[13],giveninTableI,areused

as the default case to determine the overall performance of mi-

crochannel heat sinks. The fluid properties are evaluated at the

ambienttemperature.TheresultsobtainedareshowninTableII.

Microchannel width, height, and fin thickness are optimized

in terms of channel aspect ratio and fin spacing ratio for three

different volume flow rates and Knudsen numbers in the slip

flowregion. Thecorrespondingvalues ofthetotalthermal resis-

tance, pressure drop, and entropy generation rate are tabulated.

It is observed that both channel aspect and fin spacing ratios in-

crease with the increase in volume flow rate and Knudsen num-

bers in the slip flow region. The total thermal resistance and the

pressure drop are found to decrease with the increase in volume

flow rates and increase with the decrease in Knudsen numbers.

A parametric study is carried out to investigate the depen-

dence of the optimal dimensions of the microchannel heat sink

onthethermal and hydraulicperformance. Thecurrentstudyat-

temptstofindanoptimal flowrate, channelwidth,heat sinkma-

terial, and the effect of tangential momentum and energy coeffi-

cients. Fig. 2 shows the variation of the total entropy generation

rate as a function of volume flow rate for three different values

of Knudsen numbers. This figure shows that as the Knudsen

number increases the optimal entropy generation rate decreases

Fig. 2. Effect of volume flow rate on ?

.

due to the increase in velocity slip and temperature jump at the

wall that lead to reduced heat transfer and momentum transfer

from the wall to the fluid. It also shows that the optimal flow

rate increases with the increase in Knudsen number.

The friction factor and the Nusselt number, in the laminar

flowregion,dependonthechannelaspectratio

this aspect ratio is shown in Fig. 3 for three different Knudsen

numbers in the slip flow region. The increase in aspect ratio in-

creases the cross-sectional area available for flow and the total

surface area available for convective heat transfer, which re-

duces thermal and hydraulic resistances. This figure also shows

the decrease in the optimum entropy generation rate with an in-

crease in Knudsen number.

.Theeffectof

Page 7

KHAN et al.: OPTIMIZATION OF MICROCHANNEL HEAT SINKS USING ENTROPY GENERATION MINIMIZATION METHOD 249

Fig. 3. Effect of channel aspect ratio on ?

.

Fig. 4. Effect of fin spacing ratio on ?

.

The fin spacing ratio

transfer analysis. It should be greater than 1 to ensure that there

is flow in the microchannel. The effect of this ratio is shown in

Fig. 4 for the same Knudsen numbers in the slip flow region.

For each Knudsen number, the fin spacing ratio is optimized

to get a minimum entropy generation rate. It can be observed

that the optimum fin spacing ratio is a very weak function of

the Knudsen number, however, optimal entropy generation rate

depends upon the Knudsen number and decreases with an in-

crease in the Knudsen number. It shows that lower fin spacing

ratios are appropriate in the case of microchannel heat sinks.

The total entropy generation rate

tions due to heat transfer and viscous dissipation. The thermal

conductivity of the heat sink material affects only the contribu-

tion due to heat transfer

, whereas the contribution due to

viscous dissipation

remains unchanged. Fig. 5 shows the

plays an important role in the heat

includes the contribu-

Fig. 5. Effect of heat sink material on ?

.

Fig. 6. Effect of accommodation coefficients on ?

.

effect of thermal conductivity on the

generation rate decreases with the increase in Knudsen num-

bers. This figure shows that the entropy generation rate due to

heat transfer decreases sharply from 25 to 180 W/m K and then

becomes almost constant.

The accommodation coefficients model the momentum and

energy exchange of the gas molecules impinging on the walls.

They are dependent on the specific gas and the surface quality

and are tabulated in [36]. Very low values of

creasethesliponthewallsconsiderablyevenforsmallKnudsen

number flows, due to the 2-

fectoftheseaccommodationcoefficientsisshowninFig.6.This

figure shows that for high Knudsen numbers, in the slip flow re-

gion,thereisconsiderablechangeinthetotalentropygeneration

ratebutasthe Knudsennumberdecreases, thelosses due to heat

transfer and fluid friction become negligible.

. Again the entropy

and will in-

factor in (26) and (27). The ef-

Page 8

250 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL. 32, NO. 2, JUNE 2009

IX. CONCLUSION

Based on the results of case studies and parametric optimiza-

tion, we have the following conclusions.

1) Thermal resistance and pressure drop across the mi-

crochannel decrease with an increase in volume flow rate

and increase with a decrease in Knudsen numbers in the

slip flow region.

2) Theoptimumchannelaspectandfinspacingratiosincrease

with the volume flow rate to allow the decrease in thermal

resistance and pressure drop.

3) The optimum entropy generation rate decreases with the

increase in Knudsen numbers in the slip flow region.

4) Due to slip flow and temperature jump boundary condi-

tions, fluid friction decreases and heat transfer increases in

the microchannels which decreases the total entropy gen-

eration rate.

5) A low thermal conductivity heat sink with a large number

ofmicrochannelsgivesacceptableperformanceintermsof

entropy generation rate.

6) Lower tangential momentum and energy accommodation

coefficients results in higher entropy generation rates.

REFERENCES

[1] D. B. Tuckerman and R. F. W. Pease, “High-performance heat sinking

forVLSI,”IEEEElectronDeviceLett.,vol.EDL-2,no.5,pp.126–129,

May 1981.

[2] M. E. Steinke and S. G. Kandlikar, “Single-phase heat transfer en-

hancement techniques in microchannel and minichannel flows,” in

Proc. 2nd Int. Conf. Microchannels Minichannels, Rochester, NY, Jun.

17–19, 2004, pp. 141–148.

[3] S. G. Kandlikar and W. J. Grande, “Evaluation of single phase flow

in microchannels for high flux chip cooling—Thermohydraulic perfor-

manceevaluationandfabricationtechnology,”HeatTransferEng.,vol.

25, no. 8, pp. 5–16, 2004.

[4] R. W. Knight, J. S. Goodling, and D. J. Hall, “Optimal thermal design

offorcedconvectionheatsinks-analytical,”J.Electron.Packaging,vol.

113, no. 3, pp. 313–321, 1991.

[5] R. W. Knight, D. J. Hall, J. S. Goodling, and R. C. Jaeger, “Heat

sink optimization with application to micro-channels,” IEEE Trans.

Compon. Hybrids Manuf. Technol., vol. 15, no. 5, pp. 832–842, Aug.

1992.

[6] C. Perret, C. Schaeffer, and J. Boussey, “Microchannel integrated heat

sinks in silicon technology,” in Proc. IEEE Ind. Appl. Conf., St.Louis,

MO, Oct. 12–15, 1998, vol. 2, pp. 1051–1055.

[7] C. Perret, J. Boussey, C. Schaeffer, and M. Coyaud, “Analytic mod-

eling, optimization, and realization of cooling devices in silicon tech-

nology,” IEEE Trans. Compon. Packag. Technol., vol. 23, no. 4, pp.

665–672, Dec. 2000.

[8] S. J. Kim, “Methods for thermal optimization of microchannel heat

sinks,” Heat Transfer Eng., vol. 25, no. 1, pp. 37–49, 2004.

[9] H. R. Upadhye and S. G. Kandlikar, “Optimization of microchannel

geometry for direct chip cooling using single phase heat transfer,” in

Proc. 2nd Int. Conf. Microchannels Minichannels, 2004, pp. 679–685.

[10] D. Liu and S. V. Garimella, “Analysis and optimization of the thermal

performance of microchannel heat sinks,” Int. J. Numer. Methods Heat

Fluid Flow, vol. 15, no. 1, pp. 7–26, 2005.

[11] V. Singhal, D. Liu, and S. V. Garimella, “Analysis of pumping re-

quirements for microchannel cooling systems,” in Proc. Int. Electron.

Packag.Tech.Conf.Exhib.,Maui,HI,Jul.6–11,2003,vol.2,Advances

in Electronic Packaging, pp. 473–479.

[12] S. G. Kandlikar and H. R. Upadhye, “Extending the heat flux limit

with enhanced microchannels in direct single-phase cooling of com-

puter chips,” in Proc. IEEE 21st Annu. Symp. Semiconductor Thermal

Meas. Manage., Mar. 15–17, 2005, pp. 8–15.

[13] M.B.Kleiner,S.A.Kuhn,andK.Haberger,“Highperformanceforced

air cooling scheme employing microchannel heat exchangers,” IEEE

Trans.Compon.Packag.Manuf.Technol.A,vol.18,no.4,pp.795–804,

Dec. 1995.

[14] A. Aranyosi, L. M. R. Bolle, and H. A. Buyse, “Compact air-cooled

heat sinks for power packages,” IEEE Trans. Compon. Packag. Manuf.

Technol. A, vol. 20, no. 4, pp. 442–451, Dec. 1997.

[15] C. Harris, M. Despa, and K. Kelly, “Design and fabrication of a cross

flow micro heat exchanger,” J. Microelectromech. Syst., vol. 9, pp.

502–508, 2000.

[16] S. V. Garimella and V. Singhal, “Single-phase flow and heat transport

andpumpingconsiderationsinmicrochannelheatsinks,”HeatTransfer

Eng., vol. 25, no. 1, pp. 15–25, 2004.

[17] S.P.JangandS.J.Kim,“Fluidflowandthermalcharacteristicsofami-

crochannel heat sink subject to an impinging air jet,” J. Heat Transfer,

vol. 127, no. 7, pp. 770–779, 2005.

[18] S. F. Choquette, M. Faghri, M. Charmchi, and Y. Asako, “Optimum

designofmicrochannelheatsinks,”ASMEDyn.Syst.ControlDiv.,vol.

59, pp. 115–126, 1996.

[19] W. Zhimin and C. K. Fah, “Optimum thermal design of microchannel

heat sinks,” in Proc. 1st Electron. Packag. Technol. Conf., Singapore,

Oct. 8–10, 1997, pp. 123–129.

[20] L. Meysenc, L. Saludjian, A. Bricard, S. Rael, and C. Schaeffer, “A

high heat flux IGBT micro exchanger setup,” IEEE Trans. Compon.

Packag. Manuf. Technol. A, vol. 20, no. 3, pp. 334–341, Sep. 1997.

[21] S. H. Chong, K. T. Ooi, and W. T. Wong, “Optimization of single

anddoublelayercounterflowmicrochannelheat sinks,”Appl.Thermal

Eng., vol. 22, no. 14, pp. 1569–1585, 2002.

[22] X. S. Liao, Y. Liu, Y. Q. Ning, D. M. Cheng, L. Wang, and L. J.

Wang, “The optimal design of structure parameters for microchannel

heat sink,” Proc. SPIE—Int. Soc. Opt. Eng., vol. 4914, pp. 181–186,

2002.

[23] J. H. Ryu, D. H. Choi, and S. J. Kim, “Numerical optimization of the

thermal performance of a microchannel heat sink,” Int. J. Heat Mass

Transfer, vol. 45, no. 13, pp. 2823–2827, 2002.

[24] X. Wei and Y. Joshi, “Optimization study of stacked micro-channel

heatsinksformicro-electroniccooling,”IEEETrans.Compon.Packag.

Technol., vol. 26, no. 1, pp. 55–61, Mar. 2003.

[25] G. Jean-Antoine, B. Christophe, and T. Bernard, “Extruded mi-

crochannel-structured heat exchangers,” Heat Transfer Eng., vol. 26,

no. 3, pp. 56–63, 2005.

[26] J. Y. Min, S. P. Jang, and S. J. Kim, “Effect of tip clearance on the

cooling performance of a microchannel heat sink,” Int. J. Heat Mass

Transfer, vol. 47, no. 5, pp. 1099–1103, 2004.

[27] E. R. Delsman, A. Pierik, M. H. J. M. De Croon, G. J. Kramer, and J.

C.Schouten,“Microchannelplategeometryoptimizationforevenflow

distribution at high flow rates,” Chem. Eng. Res. Design, vol. 82, no. 2,

pp. 267–273, 2004.

[28] O. Haddad, M. Abuzaid, and M. Al-Nimr, “Entropy generation due to

laminar incompressible forced convection flow through parallel-plates

microchannel,” Entropy, vol. 6, no. 5, pp. 413–426.

[29] W. A. Khan, M. M. Yovanovich, and J. R. Culham, “Analytical mod-

eling of fluid flow and heat transfer in microchannel heat sinks,” Int. J.

Heat Mass Transfer, submitted for publication.

[30] W. M. Kays and A. L. London, Compact Heat Exchangers.

York: McGraw-Hill, 1964.

[31] W. F. Stoecker, Design of Thermal Systems.

Hill, 1989.

[32] R. J. Culham and Y. S. Muzychka, “Optimization of plate fin heat

sinks using entropy generation minimization,” IEEE Trans. Compon.

Packag. Technol., vol. 24, no. 2, pp. 159–165, Jun. 2001.

[33] R. J. Culham, W. A. Khan, M. M. Yovanovich, and Y. S. Muzychka,

“The influence of material properties and spreading resistance in the

thermaldesign of plate fin heat sinks,” in Proc.35th Nat. Heat Transfer

Conf., Anaheim, CA, Jun. 10–12, 2001, pp. 240–246.

[34] W. A. Khan, J. R. Culham, and M. M. Yovanovich, “Optimization of

pin-finheatsinksusingentropygenerationminimization,”IEEETrans.

Compon. Packag. Technol., vol. 28, no. 2, pp. 247–254, Jun. 2005.

[35] W. A. Khan, J. R. Culham, and M. M. Yovanovich, “Optimal design

of tube banks in crossflow using entropy generation minimization

method,” AIAA J. Thermophys. Heat Transfer, vol. 21, no. 2, pp.

372–378, Apr./Jun. 2007.

[36] S. A. Schaaf and P. L. Chambre, Flow of Rarefied Gases.

NJ: Princeton Univ. Press, 1961.

New

New York: McGraw-

Princeton,

Page 9

KHAN et al.: OPTIMIZATION OF MICROCHANNEL HEAT SINKS USING ENTROPY GENERATION MINIMIZATION METHOD 251

Waqar Ahmed Khan received the Ph.D. degree

from the University of Waterloo, Waterloo, ON,

Canada.

Currently, heisanAssociateProfessorofMechan-

ical Engineering at the National University of Sci-

ences and Technology, Karachi, Pakistan. He has de-

velopedseveraluniqueanalyticalmodelsforthefluid

flow and heat transfer across single cylinders (cir-

cular/elliptical), tube banks and pin-fin heat sinks to

Newtonian and non-Newtonian fluids. His research

interests include modeling of forced convection heat

transfer from complex geometries, microchannel heat sinks, thermal system op-

timizationusingentropygenerationminimization,forcedandmixedconvection,

and conjugate heat transfer in air and liquid cooled applications. He has more

than 28 publications in refereed journals and international conferences.

He is a member of the American Society of Mechanical Engineers (ASME),

the American Institute of Aeronautics and Astronautics (AIAA), and the Pak-

istan engineering council.

J. Richard Culham (M’98) received the Ph.D. de-

gree from the University of Waterloo, Waterloo, ON,

Canada.

Currently, he is a Professor of Mechanical and

MechatronicsEngineeringandtheAssociateDeanof

Engineering for Research and External Partnerships

at the University of Waterloo. He is the Director

and a Founding Member of the Microelectronics

Heat Transfer Laboratory. Research interests include

modeling and characterization of contacting inter-

faces and thermal interface materials, development

of compact analytical and empirical models at micro- and nanoscales, natural

and forced convection cooling, optimization of electronics systems using

entropy generation minimization, and the characterization of thermophysical

properties in electronics and optoelectronics materials. He has more than 100

publications in refereed journals and conferences in addition to numerous

technical reports related to microelectronics cooling.

Prof. Culham is a member of the Professional Engineers Ontario.

M. Michael Yovanovich received the Sc.D. degree

from the Massachusetts Institute of Technology,

Cambridge.

Currently, he is a Distinguished Professor Emer-

itus of Mechanical Engineering at the University

of Waterloo, Waterloo, ON, Canada, and is the

Principal Scientific Advisor to the Microelectronics

Heat Transfer Laboratory. His research in the field

of thermal modeling includes analysis of complex

heat conduction problems, external and internal

natural and forced convection heat transfer from and

in complex geometries, and contact resistance theory and applications. He

has published more than 350 journal and conference papers, and numerous

technical reports, as well as three chapters in handbooks on conduction and

thermal contact resistance. He has been a consultant to several North American

nuclear, aerospace and microelectronics industries and national laboratories.

Dr. Yovanovich has received numerous awards for his teaching and his

significant contributions to science and engineering. He is the recipient of the

American Institute of Aeronautics and Astronautics (AIAA) Thermophysics

Award and the American Society of Mechanical Engineers (ASME) Heat

Transfer Award. He is a Fellow of AAAS, AIAA, and ASME.